WASHINGTON — Congress returns to work next week facing a crowded legislative agenda and one big question for President Barack Obama and Republican lawmakers: Can Washington govern? At no time since the earliest days of Obama’s presidency have conditions seemed as potentially favorable for cooperation as they do this spring, largely because both sides have reasons to want to show results.

That’s the optimistic view. Running counter is another reality. The agenda is extraordinarily challenging, and trust remains at a minimum.

The issues on the calendar — gun control, immigration and the budget — strain the coalitions of each party. At the same time, existing philosophical differences and the rigid contours of political polarization stand in the way of agreements.

Obama has spent the winter working a bifurcated strategy. One part is his so-called charm offensive, his outreach to Senate Republicans designed to create some sense of good will. Next week he will have another dinner with GOP senators, hoping to create a climate more favorable to productive negotiations.

The other part of the strategy is regular trips outside of Washington designed to rally public opinion and to ratchet up pressure on Republicans to compromise. Obama just finished such a trip, which included a rally for his gun-control proposal, which faces mounting opposition, as well as a visit to California to raise money for the 2014 elections.

Obama’s latest tactical move came Friday, with reports that he will offer Congress a fiscal blueprint that includes cuts in future spending on Medicare and Social Security. It is a budget designed to satisfy neither congressional Republicans nor his party’s left flank.

White House officials, who described the outlines of the package to reporters, heralded the budget as a signal to Republicans that Obama is serious about making concessions to produce the grand bargain that has eluded him for two years.

Obama’s willingness to cut the two big entitlements programs dates back to his negotiations with House Speaker John Boehner, R-Ohio, over the debt ceiling in the summer of 2011. At the time, he persuaded House Democratic Leader Nancy Pelosi, D-Calif., and Senate Majority Leader Harry Reid, D-Nev., to accept cuts in Social Security and Medicare, and White House officials long have claimed that the two leaders agreed to help rally their colleagues to accept them as part of a grand bargain.

Obama’s budget offer does not represent a breakthrough or even a fresh start. It is a small but perhaps symbolically important gesture. For the first time he is willing to put in writing what he has told Republicans in private negotiations, which is that he is prepared to cut both Medicare and Social Security as part of a compromise aimed at putting the country’s fiscal house in order. But, and only but, as part of a deal that includes more new revenues.

Republicans oppose any new taxes as part of a budget agreement. That was made clear again Friday morning, when Boehner dismissed the president’s forthcoming budget because it includes new taxes. Boehner’s statement was neither a surprise nor likely the last word on this.

Both the president and congressional Republicans have incentives to make the most of the time between now and the August recess to turn hopeful words into productive results. There might be few better opportunities for the rest of Obama’s presidency than now.

The president knows he has limited time to produce results on big issues before the 2014 elections and eventual lame duck status get in the way of serious bargaining.

Republicans, with their approval ratings still low and having lost the popular vote in five of the past six elections, have more than enough reasons to show they are more than the party of “no.”

However, the prospects for legislative deal-making between now and the summer recess remain questionable.

Obama’s gun-control package has run into opposition, despite public support for many of the measures he has advocated.

Immigration is another matter. Nowhere do Republicans have greater self-interest to enact legislation than on immigration. The party’s deficit with Latino voters has prompted many Republican leaders to embrace a path to citizenship or legal status as part of comprehensive reform.

Despite the obvious political incentives, Republicans return next week still in search of a legislative solution that would take the issue off the table politically without rupturing the party’s conservative coalition.

Republicans see the need for a new position on immigration. A Republican organization, Resurgent Republic, reported on four focus groups that showed GOP primary voters are more open to immigration reform with a path to some kind of legal status, but only with stringent conditions.

Meanwhile, there is little evidence that opinion among conservatives has shifted dramatically on the issue.

That is one reason why Sen. Marco Rubio, R-Fla., who is part of the group trying to reach a bipartisan compromise, has warned Democrats not to try to jam the legislation through. Republican leaders who want to see the reforms think they need time to bring their base along. Immigration might offer the best current hope for bipartisan agreement, but not quite yet.

That leaves the budget as the biggest challenge, the most contentious issue and the most significant prize. Sen. Mark Warner, D-Va., speaking recently at a breakfast hosted by Bloomberg News, said there is nothing more important for Washington politicians than breaking the fever of angst and dysfunction that has been the norm to prove to voters that the system is not irretrievably broken. He said the “metric for that, whether we like it or not … is the debt, deficit [and] budget issues.”

Between now and late summer, politicians will provide the answer to the question of whether Washington can govern. With the economy struggling and the impact of the across-the-board spending cuts starting to bite, that’s more than an idle question.

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